Efficient Beltrami Flow Using a Short Time Kernel

We introduce a short time kernel for the Beltrami image enhancing flow. The flow is implemented by 'convolving' the image with a space dependent kernel in a similar fashion to the implementation of the heat equation by a convolution with a gaussian kernel. The expression for the kernel shows, yet again, the connection between the Beltrami flow and the Bilateral filter. The kernel is calculated by measuring distances on the image manifold by an efficient variation of the fast marching method. The kernel, thus obtained, can be used for arbitrary large time steps in order to produce adaptive smoothing and/or a new scale-space. We apply it to gray scale and color images to demonstrate its flow like behavior.

[1]  Danny Barash,et al.  A Fundamental Relationship between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Diffusion Equation , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[2]  R. Kimmel,et al.  An efficient solution to the eikonal equation on parametric manifolds , 2004 .

[3]  F. Mémoli,et al.  Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces: 730 , 2001 .

[4]  Ron Kimmel,et al.  From High Energy Physics to Low Level Vision , 1997, Scale-Space.

[5]  Alexander Vladimirsky,et al.  Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms , 2003, SIAM J. Numer. Anal..

[6]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[8]  Alfred M. Bruckstein,et al.  Diffusions and Confusions in Signal and Image Processing , 2001, Journal of Mathematical Imaging and Vision.

[9]  Michael Elad,et al.  On the bilateral filter and ways to improve it , 2002 .

[10]  J. Sethian,et al.  Ordered upwind methods for static Hamilton–Jacobi equations , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Nir A. Sochen Stochastic Processes in Vision: From Langevin to Beltrami , 2001, ICCV.

[12]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Ron Kimmel,et al.  Image Processing via the Beltrami Operator , 1998, ACCV.

[14]  Nir Sochen,et al.  Stochastic processes in vision: from Langevin to Beltrami , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[15]  Roberto Manduchi,et al.  Bilateral filtering for gray and color images , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[16]  Michael Elad,et al.  On the origin of the bilateral filter and ways to improve it , 2002, IEEE Trans. Image Process..

[17]  Yehoshua Y. Zeevi,et al.  Representation of colored images by manifolds embedded in higher dimensional non-Euclidean space , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).